Title Computing the genus of an algebraic
curve. |
Version
française |

**Location**

INRIA, Project
CAFÉ

BP 93, 06902 France

**Information**

**Description**

The genus of an algebraic curve is a fundamental quantity that several
algorithms in cryptography or computer algebra depend on. Computing it,
which is often a preliminary step for other algorithms, remains difficult
for general curves. Last year, an intern implemented
the new method [3], which is based on Hurwitz' formula and uses an ordinary
linear differential operator associated with the curve. An experimental
method based on a differential system of the form *Y' = A(x) Y *and
the reduction of [1] to compute its exponents was also implemented [4]
and found to be faster that the method of [3], but still more expensive
than local desingularisation. A new improvement of that method that first
computes a *suitable basis* of the associated algebraic function field
using the algorithm of [2] and then the corresponding *explicitely regular*
differential system *W' = B(x) W* to compute the ramification
indices is to be implemented and compared to the previous work. This method
is expected to be competitive with the fastest known algorithms, and to
be applicable to other computations, such as absolute factorization (decomposition
of the curve) and computation of the Galois group of the polynomial defining
the curve.

It is possible to continue this work towards a doctoral thesis around
the above applications, together with a generalization to hypersurfaces
in higher dimensions using systems of linear *partial* differential
equations rather than ordinary equations.

[1] S.Abramov and M.Bronstein (2001): *On
Solutions of Linear Functional Systems,* in Proceedings of ISSAC'2001,
London (Ontario), ACM Press, pp.1-6.

[2] M.Bronstein (1998): *The
Lazy Hermite Reduction,* INRIA Research Report 3562.

[3] O.Cormier, M.Singer, B.Trager and F.Ulmer (2001): *Linear
Differential Operators for Polynomial Equations,* manuscript submitted
to the Journal of Symbolic Computation.

[4] E.Dottax (2001): *Calcul du genre d'une
courbe algébrique*, Rapport de stage INRIA.

**Tools**

Unix workstation, a computer algebra system (Axiom or Maple), then Aldor programming language.

**Duration**

3 - 4 months, possible prolongation as a doctoral thesis if desired.